How many different triangles can be constructed with toothpicks by conncting the toothpicks only at their ends if each triangle can contain at most five toothpicks per side? I know the answer is 22 but I have no idea how they came up with this answer. Any ideas?

To find the number of different triangles that can be constructed with toothpicks, we can break down the problem into smaller cases and count the possibilities for each case.

First, let's consider triangles with sides of length one toothpick. In this case, we can only have an equilateral triangle. So, there is only one possibility.

Next, let's consider triangles with sides of length two toothpicks. These triangles can either be isosceles or scalene. Let's count the possibilities for each:

1. Isosceles triangle: If two sides are equal, we can have two scenarios: both sides connected directly to form a line, or the two sides connected at one end and a third toothpick connected to the other ends. So, there are two possibilities.

2. Scalene triangle: If all three sides are of different lengths, we can have three scenarios: all three toothpicks connected at one end, two toothpicks connected at one end and the third toothpick connected to the other ends, or all three toothpicks connected at different ends. So, there are three possibilities.

Now, let's consider triangles with sides of length three toothpicks:

1. Equilateral triangle: Just like in the case of one toothpick per side, there is only one possibility.

2. Isosceles triangle: If two sides are equal, we have two scenarios: both sides connected directly to form a line, or the two sides connected at one end and a third toothpick connected to the other ends. So, there are two possibilities.

3. Scalene triangle: If all three sides are of different lengths, we again have three scenarios: all three toothpicks connected at one end, two toothpicks connected at one end and the third toothpick connected to the other ends, or all three toothpicks connected at different ends. So, there are three possibilities.

Finally, let's consider triangles with sides of length four and five toothpicks:

For both four and five toothpicks per side, we can only have one possibility, which is connecting all the toothpicks at different ends.

Summing up all the possibilities for each case, we have:

1 + 2 + 3 + 1 + 2 + 3 + 1 + 1 = 14

Therefore, there are a total of 14 different triangles that can be constructed with toothpicks using the given conditions. It seems that the given answer of 22 is incorrect based on the analysis above.